2.3 Riesz representation theorem

The Riesz representation theorem is a cornerstone of functional analysis, bridging linear functionals and Hilbert space elements. It states that every bounded linear functional on a Hilbert space can be uniquely represented as an inner product with a fixed vector.

This theorem has far-reaching implications in spectral theory, quantum mechanics, and operator analysis. It provides a concrete way to study abstract linear functionals, enabling deeper insights into the structure of Hilbert spaces and their applications in various fields of mathematics and physics.

Statement of theorem

• Riesz representation theorem establishes a correspondence between linear functionals and elements of a Hilbert space
• Provides a powerful tool for analyzing linear functionals in Spectral Theory by representing them as inner products
• Bridges the gap between abstract linear functionals and concrete vector representations in Hilbert spaces

Hilbert space formulation

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• States that every bounded linear functional $L$ on a Hilbert space $H$ can be represented uniquely as an inner product
• Representation takes the form $L(x) = \langle x, y \rangle$ for some fixed $y \in H$
• Norm of the functional equals the norm of the representing vector: $\|L\| = \|y\|$
• Applies to both real and complex Hilbert spaces

Banach space formulation

• Extends the theorem to certain Banach spaces (reflexive spaces)
• Identifies the dual space of a reflexive Banach space with the original space
• Uses the notion of conjugate linear functionals instead of inner products
• Requires additional conditions on the Banach space (uniformly convex)

Key concepts

• Linear functionals, dual spaces, and inner products form the foundation of the Riesz representation theorem
• Understanding these concepts aids in grasping the theorem's significance in Spectral Theory
• Provides a framework for analyzing linear operators and their properties

Linear functionals

• Defined as linear maps from a vector space to its underlying scalar field
• Preserve vector addition and scalar multiplication
• Can be represented as $L: V \rightarrow \mathbb{F}$ where $V$ is a vector space and $\mathbb{F}$ is the scalar field
• Play a crucial role in functional analysis and operator theory

Dual spaces

• Consist of all continuous linear functionals on a given vector space
• Denoted as $V^*$ for a vector space $V$
• Form a vector space themselves with pointwise addition and scalar multiplication
• Provide insight into the structure and properties of the original space

Inner products

• Generalize the notion of dot product to abstract vector spaces
• Define a bilinear (or sesquilinear) map $\langle \cdot, \cdot \rangle: V \times V \rightarrow \mathbb{F}$
• Satisfy properties of conjugate symmetry, linearity, and positive-definiteness
• Enable the definition of orthogonality and norms in Hilbert spaces

Proof outline

• Proof of the Riesz representation theorem involves three main steps
• Demonstrates the power of Hilbert space geometry in functional analysis
• Utilizes key properties of Hilbert spaces (completeness, orthogonality)

Existence of representation

• Construct a closed subspace $M = \{x \in H : L(x) = 0\}$
• Use the orthogonal decomposition theorem to write $H = M \oplus M^\perp$
• Show that $M^\perp$ is one-dimensional and spanned by a vector $y$
• Prove that $L(x) = \langle x, cy \rangle$ for some scalar $c$

Uniqueness of representation

• Assume two representations $L(x) = \langle x, y_1 \rangle = \langle x, y_2 \rangle$
• Show that $\langle x, y_1 - y_2 \rangle = 0$ for all $x \in H$
• Conclude $y_1 = y_2$ using the positive-definiteness of inner products

Norm preservation

• Prove $\|L\| = \sup_{\|x\| \leq 1} |L(x)| = \sup_{\|x\| \leq 1} |\langle x, y \rangle|$
• Use Cauchy-Schwarz inequality to show $|\langle x, y \rangle| \leq \|x\| \|y\|$
• Demonstrate that equality holds when $x$ is a scalar multiple of $y$

Applications

• Riesz representation theorem has wide-ranging applications in mathematics and physics
• Provides a powerful tool for analyzing linear operators and functionals
• Connects abstract functional analysis to concrete physical interpretations

Spectral theory

• Enables the study of spectral properties of linear operators
• Facilitates the representation of compact operators as infinite sums
• Aids in the analysis of self-adjoint operators and their eigenvalue decompositions

Functional analysis

• Provides a concrete realization of dual spaces in Hilbert spaces
• Simplifies the study of weak topologies and weak convergence
• Facilitates the development of operator theory and Banach algebra theory

Quantum mechanics

• Allows representation of quantum observables as self-adjoint operators
• Enables the formulation of the measurement postulate using projection operators
• Facilitates the study of expectation values and uncertainty relations

Extensions and variants

• Riesz representation theorem can be extended and modified for different settings
• Adaptations allow for application to a broader range of mathematical spaces
• Highlights the versatility and power of the theorem in various contexts

Complex vs real spaces

• Theorem applies to both real and complex Hilbert spaces
• Complex case requires conjugate linearity in the second argument of the inner product
• Real case simplifies some aspects but may lose certain algebraic properties

Reflexive spaces

• Extends the theorem to certain Banach spaces beyond Hilbert spaces
• Requires the space to be reflexive (isomorphic to its double dual)
• Provides a weaker form of representation using duality pairings instead of inner products

Weak topologies

• Theorem plays a crucial role in understanding weak and weak* topologies
• Enables characterization of weakly convergent sequences in Hilbert spaces
• Facilitates the study of weak compactness and its applications

Historical context

• Riesz representation theorem emerged during the development of functional analysis
• Played a crucial role in formalizing the study of infinite-dimensional spaces
• Influenced the direction of research in operator theory and spectral analysis

Development of functional analysis

• Arose from the need to study integral and differential equations
• Emerged in the early 20th century as a distinct field of mathematics
• Integrated concepts from analysis, algebra, and geometry

Contributions of Frigyes Riesz

• Hungarian mathematician who made significant contributions to functional analysis
• Formulated the Riesz representation theorem in 1907
• Also developed the spectral theorem for compact operators

Connections to other theorems

• Riesz representation theorem relates to several other fundamental results in functional analysis
• Forms part of a network of theorems that characterize linear functionals and operators
• Provides a foundation for more advanced results in spectral theory

Hahn-Banach theorem

• Extends bounded linear functionals from subspaces to the whole space
• Complements Riesz representation by addressing existence of linear functionals
• Applies to general normed spaces, not just Hilbert spaces

Lax-Milgram theorem

• Generalizes Riesz representation to bounded bilinear forms
• Provides conditions for existence and uniqueness of weak solutions to PDEs
• Crucial in the study of variational problems and finite element methods

Spectral theorem

• Characterizes normal operators on Hilbert spaces
• Utilizes the Riesz representation to construct spectral measures
• Fundamental in quantum mechanics and operator algebra theory

Examples and counterexamples

• Illustrating the Riesz representation theorem with concrete examples aids understanding
• Counterexamples highlight the limitations and necessary conditions for the theorem
• Provides insight into the theorem's applicability in various mathematical settings

Finite-dimensional spaces

• Theorem simplifies to the correspondence between vectors and linear functionals
• Representation can be explicitly constructed using the standard basis
• Dual space is isomorphic to the original space ($\mathbb{R}^n \cong (\mathbb{R}^n)^*$)

L^p spaces

• Theorem applies directly to $L^2$ spaces, which are Hilbert spaces
• For $p \neq 2$, $L^p$ spaces are not Hilbert spaces but may be reflexive
• Representation in $L^p$ involves the Hölder conjugate $q$ where $\frac{1}{p} + \frac{1}{q} = 1$

Non-reflexive spaces

• Spaces like $L^1$ and $L^\infty$ do not satisfy the conditions of the theorem
• Dual of $L^1$ is $L^\infty$, but the converse does not hold
• Illustrates the importance of reflexivity in extending the theorem to Banach spaces

Implications for operators

• Riesz representation theorem has profound implications for the study of linear operators
• Provides a framework for analyzing various classes of operators in Hilbert spaces
• Facilitates the development of spectral theory and functional calculus

Adjoint operators

• Theorem enables the definition of adjoint operators in Hilbert spaces
• For an operator $T$, its adjoint $T^*$ satisfies $\langle Tx, y \rangle = \langle x, T^*y \rangle$
• Crucial in the study of self-adjoint and normal operators

Compact operators

• Representation theorem aids in characterizing compact operators
• Allows decomposition of compact operators into sum of rank-one operators
• Facilitates the study of spectral properties of compact operators

Self-adjoint operators

• Theorem provides a powerful tool for analyzing self-adjoint operators
• Enables spectral decomposition and functional calculus for self-adjoint operators
• Fundamental in quantum mechanics for representing physical observables